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Dimensional Enhancement Via Supersymmetry Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2011, Article ID 259089, 45 pages doi:10.1155/2011/259089 Research Article Dimensional Enhancement via Supersymmetry M. G. Faux,1 K. M. Iga,2 and G. D. Landweber3 1 Department of Physics, State University of New York, Oneonta, NY 13820, USA 2 Natural Science Division, Pepperdine University, Malibu, CA 90263, USA 3 Department of Mathematics, Bard College, Annandale-on-Hudson, NY 12504-5000, USA Correspondence should be addressed to M. G. Faux, [email protected] Received 3 March 2011; Revised 24 May 2011; Accepted 24 May 2011 Academic Editor: Yao-Zhong Zhang Copyright q 2011 M. G. Faux et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited, and utilized in this paper, which indicate which subset of one-dimensional supersymmetric models describes “shadows” of higher-dimensional models. This formalism delineates that minority of one-dimensional supersymmetric models which can “enhance” to accommodate extra dimensions. As a consistency test, we use our formalism to reproduce well-known conclusions about supersymmetric field theories using one- dimensional reasoning exclusively. And we introduce the notion of “phantoms” which usefully accommodate higher-dimensional gauge invariance in the context of shadow multiplets in supersymmetric quantum mechanics. 1. Introduction Supersymmetry 1–5 imposes increasingly rigid constraints on the construction of quantum field theories 6–11 as the number of space-time dimensions increases. Thus, there are fewer supersymmetric models in six dimensions than there are in four, and yet fewer in ten dimensions 12. In eleven dimensions there seems to be a unique possibility 13,at least on-shell. Anomaly freedom imposes seemingly distinct algebraic constraints which make this situation even more interesting. However, the off-shell representation theory for supersymmetry is well understood only for relatively few supersymmetries, and remains a mysterious subject in contexts of special interest, such as N 4 Super Yang Mills theory, and the four ten-dimensional supergravity theories 14. Many lower-dimensional models can be obtained from higher-dimensional models by dimensional reduction 15, 16. Thus, a subset of lower-dimensional supersymmetric 2 Advances in Mathematical Physics theories derives from the landscape of possible ways that extra dimensions can be removed. But most lower-dimensional theories do not seem to be obtainable from higher-dimensional theories by such a process; they seem to exist only in lower-dimensions. We refer to a lower- dimensional model obtained by dimensional reduction of a higher-dimensional model as the “shadow” of the higher-dimensional model. So we could rephrase our comment above by saying that not all lower dimensional supersymmetric theories may be interpreted as shadows. It is a straightforward process to construct a shadow theory from a given higher- dimensional theory. But it is a more subtle proposition to construct a higher-dimensional supersymmetric model from a lower-dimensional model, or to determine whether a lower dimensional model actually does describe a shadow, especially of a higher-dimensional theory which is also supersymmetric. We have found resident within lower-dimensional supersymmetry an algebraic key which provides access to this information. A primary purpose of this paper is to explain this. It is especially interesting to consider reduction to one time-like dimension, by switching off the dependence of all fields on all of the spatial coordinates. Such a process reduces quantum field theory to quantum mechanics. Upon making such a reduction, in- formation regarding the spin representation content of the component fields is replaced with R-charge assignments. But it is not obvious whether the full higher-dimensional field content, or the fact that the one-dimensional model can be obtained in this way, is accessible information given the one-dimensional theory alone. As it turns out, this infor- mation lies encoded within the extended one-dimensional supersymmetry transformation rules. We refer to the process of restructuring a one-dimensional theory so that fields depend also on extra dimensions in a way consistent with covariant spin1,D− 1 assignments and other structures, such as higher-dimensional supersymmetries, as “dimensional enhance- ment”. This process describes the reverse of dimensional reduction. We like to envision this in terms of the relationship between a higher-dimensional “ambient” theory, and the restriction to a zero-brane embedded in the larger space. A supersymmetric quantum mechanics then describes the “worldline” physics on the zero-brane. And the question as to whether this worldline physics “enhances” to an ambient space-time field theory is the reverse of viewing the worldline physics as the restriction of a target-space theory to the zero-brane. If the particular supersymmetric quantum mechanics obtained by restriction of a given theory to a zero-brane depended on the particular spin1,D − 1-frame described by that zero-brane, the higher-dimensional theory would not respect spin1,D − 1- invariance. Thus, if a one-dimensional theory enhances into a spin1,D− 1-invariant higher- dimensional theory, then the higher-dimensional theory obtained in this way should be agnostic regarding the presence or absence of an actual zero-brane on which such a one- dimensional theory might live. This observation, in conjunction with the requirement of higher-dimensional supersymmetry, provides the requisite constraint needed to resolve the enhancement question. In particular, by imposing spin1,D− 1-invariance on the enhanced supercharge operator, we are able to complete the ambient field-theoretic supercharge operator given merely the “time-like” restrictions of this operator. We find this interesting and surprising. The proposition that one can systematically delineate those one-dimensional theories which can enhance to higher-dimensions, and also discern how the higher-dimensional spin structures may be switched back on, is empowered by the fact that the representation theory of one-dimensional supersymmetry is relatively tame when compared with the Advances in Mathematical Physics 3 representation theory of higher-dimensional supersymmetry, for a variety of reasons. This enables the prospect of disconnecting the problem of spin assignments from the problem of classifying and enumerating supersymmetry representations, allowing these concerns to be addressed separately, and then merged together afterwards. With this motivation in mind, we have been developing a mathematical context for the representation theory of one- dimensional supersymmetry, also with other collaborators. The primary purpose of this paper is to demonstrate that the landscape of supersym- metry representation theory in any number of space-time dimensions resides fully-encoded within the seemingly-restricted regime of one-dimensional worldline supersymmetry representation theory. We find this result remarkable, compelling, and noteworthy, regardless of how complicated it may prove to algorithmically “extract” this information. But we demonstrate below that algorithms to perform such extractions do exist. In fact, we present explicit examples of algorithms which delineate one-dimensional models which are shadows of higher-dimensional models from those which are not. We do not purport that our algorithms are optimized. And we view this paper as a plateau from which more efficient algorithms could be developed. A cursory accounting of the complexity of the general problem is addressed in Section 5. In a sequence of papers 17–23, we have explored the connection between repre- sentations of supersymmetry and aspects of graph theory. We have shown that elements of a wide and physically relevant class of one-dimensional supermultiplets with vanishing central charge are equivalent to specific bipartite graphs which we call Adinkras; all of the salient algebraic features of the multiplets translate into restrictive and defining features of these objects. A systematic enumeration of those graphs meeting the requisite criteria would thereby supply means for a corresponding enumeration of representations of supersymmetry. In 24, 25, we have developed the paradigm further, explaining how, in the case of N-extended supersymmetry, the topology of all connected Adinkras are specified by quotients of N-dimensional cubes, and how the quotient groups are equivalent to doubly- even linear binary block codes. Thus, the classification of connected Adinkras is related to the classification of such codes. In this way we have discovered an interesting connection between supersymmetry representation theory and coding theory 26–28. All of this is part of an active endeavor aimed at delineating a mathematically-rigorous representation theory in one-dimension. In this paper we use the language of Adinkras, in a way which does not presuppose a deep familiarity with this topic. We have included Appendix B as a brief and superficial primer, which should enable the reader to appreciate the entirety of this paper self- consistently. Further information
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